Abstract
Achromatic focusing systems for hard X-rays are examined which consist of a refractive lens paired with a diffractive lens. Compared with previous analyses, we take into account the behaviour of thick refractive lenses, such as compound refractive lenses and waveguide gradient index refractive lenses, in which both the focal length and the position of the principal planes vary with wavelength. Achromatic systems formed by the combination of such a thick refractive lens with a multilayer Laue lens are found that can operate at a focusing resolution of about 3 nm, over a relative bandwidth of about 1%. With the appropriate distance between the refractive and diffractive lenses, apochromatic systems can also be found, which operate over relative bandwidth greater than 10%. These systems can be used to focus short pulses without distorting them in time by more than several attoseconds. Such systems are suitable for high-flux scanning microscopy and for creating high intensities from attosecond X-ray pulses.